Class 11th Maths  Chapter 5 Quadratic equation handwritten notes.



In this article we provide you the most important quadratic equation handwritten notes which is very important to revision the things that we had solve in the first time so we provide the most important or written by the top facilities .


❄️Notes on Quadratic Equations⛄️



In order to solve a quadratic equation of the form ax2 + bx + c, we first need to calculate the discriminant with the help of the formula D = b2 – 4ac.


The solution of the quadratic equation ax2 + bx + c= 0 is given by x = [-b ± √ b2 – 4ac] / 2a


If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then we have the following results for the sum and product of roots:


α + β = -b/a


α.β = c/a


α – β = √D/a


It is not possible for a quadratic equation to have three different roots and if in any case it happens, then the equation becomes an identity.


Nature of Roots:


Consider an equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then we have the following cases:


D > 0 iff the roots are real and distinct i.e. the roots are unequal


D = 0 iff the roots are real and coincident i.e. equal


D < 0 iffthe roots are imaginary


The imaginary roots always occur in pairs i.e. if a+ib is one root of a quadratic equation, then the other root must be the conjugate i.e. a-ib, where a, b ∈ R and i = √-1.


Consider an equation ax2 + bx + c = 0, where a, b and c ∈Q and a ≠ 0, then


If D > 0 and is also a perfect square then the roots are rational and unequal.


If α = p + √q is a root of the equation, where ‘p’ is rational and √q is a surd, then the other root must be the conjugate of it i.e. β = p - √q and vice versa.


If the roots of the quadratic equation are known, then the quadratic equation may be constructed with the help of the formula

x2 – (Sum of roots)x + (Product of roots) = 0.


So if α and β are the roots of equation then the quadratic equation is


x2 – (α + β)x + α β = 0


For the quadratic expressiony = ax2 + bx + c, where a, b, c ∈ R and a ≠ 0, then the graph between x and y is always a parabola.


If a > 0, then the shape of the parabola is concave upwards


If a < 0, then the shape of the parabola is concave upwards


Inequalities of the form P(x)/ Q(x) > 0 can be easily solved by the method of intervals of number line rule.


The maximum and minimum values of the expression y = ax2 + bx + c occur at the point x = -b/2a depending on whether a > 0 or a< 0.


 y ∈[(4ac-b2) / 4a, ∞] if a > 0


 If a < 0, then y ∈ [-∞, (4ac-b2) / 4a]


The quadratic function of the form f(x, y) = ax2+by2 + 2hxy + 2gx + 2fy + c = 0 can be resolved into two linear factors provided it satisfies the following condition: abc + 2fgh –af2 – bg2 – ch2 = 0


In general, if α1,α2, α3, …… ,αn are the roots of the equation


f(x) = a0xn +a1xn-1 + a2xn-2 + ……. + an-1x + an, then


1.Σα1 = - a1/a0


2.Σ α1α2 = a2/a0


3.Σ α1α2α3 = - a3/a0


  Σ α1α2α3 ……αn= (-1)n an/a0


Every equation of nth degree has exactly n roots (n ≥1) and if it has more than n roots then the equation becomes an identity.


If there are two real numbers ‘a’ and ‘b’ such that f(a) and f(b) are of opposite signs, then f(x) = 0 must have at least one real root between ‘a’ and ‘b’.


Every equation f(x) = 0 of odd degree has at least one real root of a sign opposite to that of its last term.